2014-11-09 06:15:30 +00:00
|
|
|
|
import data.nat.basic data.prod
|
|
|
|
|
|
open prod
|
|
|
|
|
|
|
|
|
|
|
|
namespace nat
|
2014-12-03 18:39:22 +00:00
|
|
|
|
namespace manual
|
2014-11-09 06:15:30 +00:00
|
|
|
|
definition brec_on {C : nat → Type} (n : nat) (F : Π (n : nat), @below C n → C n) : C n :=
|
|
|
|
|
|
have general : C n × @below C n, from
|
|
|
|
|
|
rec_on n
|
|
|
|
|
|
(pair (F zero unit.star) unit.star)
|
|
|
|
|
|
(λ (n₁ : nat) (r₁ : C n₁ × @below C n₁),
|
|
|
|
|
|
have b : @below C (succ n₁), from
|
|
|
|
|
|
r₁,
|
|
|
|
|
|
have c : C (succ n₁), from
|
|
|
|
|
|
F (succ n₁) b,
|
|
|
|
|
|
pair c b),
|
|
|
|
|
|
pr₁ general
|
2014-12-03 18:39:22 +00:00
|
|
|
|
end manual
|
2014-11-09 06:15:30 +00:00
|
|
|
|
|
|
|
|
|
|
definition fib (n : nat) :=
|
|
|
|
|
|
brec_on n (λ (n : nat),
|
|
|
|
|
|
cases_on n
|
|
|
|
|
|
(λ (b₀ : below zero), succ zero)
|
|
|
|
|
|
(λ (n₁ : nat), cases_on n₁
|
|
|
|
|
|
(λ b₁ : below (succ zero), succ zero)
|
|
|
|
|
|
(λ (n₂ : nat) (b₂ : below (succ (succ n₂))), pr₁ b₂ + pr₁ (pr₂ b₂))))
|
|
|
|
|
|
|
|
|
|
|
|
theorem fib_0 : fib 0 = 1 :=
|
|
|
|
|
|
rfl
|
|
|
|
|
|
|
|
|
|
|
|
theorem fib_1 : fib 1 = 1 :=
|
|
|
|
|
|
rfl
|
|
|
|
|
|
|
|
|
|
|
|
theorem fib_s_s (n : nat) : fib (succ (succ n)) = fib (succ n) + fib n :=
|
|
|
|
|
|
rfl
|
|
|
|
|
|
|
|
|
|
|
|
example : fib 5 = 8 :=
|
|
|
|
|
|
rfl
|
|
|
|
|
|
|
|
|
|
|
|
example : fib 9 = 55 :=
|
|
|
|
|
|
rfl
|
|
|
|
|
|
end nat
|
